Lucy has funds that she wants to invest in bonds, and she just noticed in the financial pages of the local newspaper that she can buy a $1,000 par value bond for $1,200. The coupon rate is 10% (with annual payments), and there are 10 years before the bond will mature and pay off its $1,000 par value. Lucy should buy the bond if her required return on bonds is单项选择题
A
A. 12%
B
B. 10%
C
C. 9%
D
D. 7%

Question

Lucy has funds that she wants to invest in bonds, and she just noticed in the financial pages of the local newspaper that she can buy a $1,000 par value bond for $1,200. The coupon rate is 10% (with annual payments), and there are 10 years before the bond will mature and pay off its $1,000 par value. Lucy should buy the bond if her required return on bonds is单项选择题

A

A. 12%

B

B. 10%

C

C. 9%

D

D. 7%

BlackTom AI · Accepted Answer

The yield to maturity (YTM) is the discount rate r that makes the present value of the bond’s cash flows equal to its market price. For this bond: coupon C = 0.10 × $1,000 = $100, face value FV = $1,000, n = 10 years, and market price P = $1,200. The bond price formula:
P = C × [1 − (1 + r)^−n]/r + FV × (1 + r)^−n.

Compute PV at the candidate required returns to see whether Lucy would buy (she should buy if the bond’s YTM ≥ her required return).

1) Evaluate at r = 7%:
(1.07)^10 ≈ 1.967151 → (1.07)^−10 ≈ 0.50835.
Annuity factor = (1 − 0.50835)/0.07 = 0.49165/0.07 = 7.02357.
PV(coupons) = 100 × 7.02357 = $702.36.
PV(par) = 1000 × 0.50835 = $508.35.
Total PV ≈ 702.36 + 508.35 = $1,210.71 (slightly above $1,200).

2) Evaluate at r = 9%:
(1.09)^10 ≈ 2.36736 → (1.09)^−10 ≈ 0.42241.
Annuity factor = (1 − 0.42241)/0.09 = 0.57759/0.09 = 6.4177.
PV(coupons) = 100 × 6.4177 = $641.77.
PV(par) = 1000 × 0.42241 = $422.41.
Total PV ≈ 641.77 + 422.41 = $1,064.18 (well below $1,200).

Quick checks: at r = 10% the price would be par ($1,000), and at r = 12% even lower. The actual YTM is between 7% and 9]; interpolating between 7% (PV ≈ $1,210.71) and 9% (PV ≈ $1,064.18) gives YTM ≈ 7.3% (so the bond yields about 7.3%).

Interpretation: Lucy should buy only if her required return is less than or equal to the bond’s YTM (because then the bond’s return meets her requirement). Among the choices, only 7% is less than the bond’s YTM (~7.3%). Therefore she should buy the bond if her required return is 7%.

Why other options are wrong:
- 12% and 10%: required returns exceed the bond’s YTM, so the bond would provide less than she requires.
- 9%: still above the bond’s YTM (~7.3%), so unacceptable.